In this lesson, we learn how to interpret limits whose values are unbounded, or infinite.
We’ll look at a few examples to show you how to compute these infinite limits.
What is an Infinite Limit?
In calculus, a limit is the y-value to which a function approaches as the x-values get closer to some specified number. But what happens where the function doesn’t approach any particular y-value? In that case, we often say that the limit does not exist. However, in some situations, a more descriptive answer can be given. We say that f approaches ∞ as x approaches a number c if the values of f(x) become larger and larger without bound as x approaches c.
The notation for this situation is:
On the other hand, we say that f approaches -∞ as x approaches c if the values of f(x) become unbounded in the negative sense (larger and larger negative values) as x approaches c. The notation is similar.
|In either situation, the function is said to have an infinite limit at the number x = c.The presence of an infinite limit in a function actually indicates that the function ‘blows up.’ You may have heard that nothing can travel faster than the speed of light.
Well, this is true in part because the function that measures the mass and energy of an object with respect to its velocity v has an infinite limit at the number v = c, where c in this case is the speed of light! The infinite limit tells us that mass/energy actually becomes unbounded as velocity approaches c; hence, no physical object can reach that speed.Now, let’s look at some examples.
A Case Study: 1/x
Consider the function f(x) = 1/x. What happens to the y-values when x approaches 0? The graph shows a vertical asymptote at x = 0, which is a good indication that we’re dealing with an infinite limit.
The function seems to shoot off the top of the graph when x approaches 0 from the right; that means the limit of f is ∞ as x → 0 from the right. The graph drops off the bottom of the graph when x approaches 0 from the left, so the limit of f is -∞ as x → 0 from the left.Let’s build a table of values to illustrate this behavior more concretely.
Notice how the y-values shown in the second column continue to increase without bound (100; 1,000,000; etc.), and the y-values shown in the fourth column decrease without bound (-100; -1,000,000; etc.). Some of the points in the table are plotted on the graph in the illustration.
Trace with your finger on the graph starting at the point (1,1). As you trace to the left, always staying on the graph, you eventually hit the top. The graph actually goes upward beyond what’s shown, and never stops! This is the essence of an infinite limit.
According to the table (or the graph), we conclude that: